Arithmetic Progression

Sequence: A list of numbers having specific relation between the consecutive terms is generally called a sequence.

e.g. 1, 3, 5, 7,……… (next term to a term is obtained by adding 2 with it)

& 2, 6, 18, 54,…….( next term to a term is obtained by multiplying 3 with it)

Arithmetic Progression: If various terms of a sequence are formed by adding a fixed number to the previous term or the difference between two successive terms is a fixed number, then the sequence is called AP.

e.g.1) 2, 4, 6, 8, ……… the sequence of even numbers is an example of AP

2) 5, 10, 15, 20, 25….. In this each term is obtained by adding 5 to

the preceding term except first term.

Illustrative example for A.P.

=d,where d=1

a a+d a+2d

a+3d………………

The general form of an Arithmetic Progression is

a , a +d , a + 2d , a + 3d ………………, a + (n-1)d

Where ‘a’ is first term and

‘d’ is called common difference.

Common Difference - The fixed number which is obtained by subtracting any term of AP from its previous term.

If we take First term of an AP as a and Common Difference as d,

Then, nth term of that AP will be

An = a + (n-1)d

3, 7, 11, 15, 19 …

Notice in this sequence that if we find the difference between any term and the term before it we always get 4. 4 is then called the common difference and is denoted with the letter d.

d =4

To get to the next term in the sequence we would add 4 so a recursive formula for this sequence is:

41 nn aa

The first term in the sequence would be a1 which is sometimes just written as a.

a =3

3, 7, 11, 15, 19 …

+4 +4 +4 +4

Each time you want another term in the sequence you’d add d. This would mean the second term was the first term plus d. The third term is the first term plus d plus d (added twice). The fourth term is the first term plus d plus d plus d (added three times). So you can see to get the nth term we’d take the first term and add d (n - 1) times.

d =4

dnaan 1

Try this to get the 5th term.

a =3

1916341535 a

Let’s see an example!!

Let a=2, d=2, n=12,find An

An=a+(n-1)d

=2+(12-1)2 =2+(11)2 =2+22 Therefore, An=24

Hence solved.

To check that a given term is in A.P. or not.

2, 6, 10, 14….Here first term a = 2,

find differences in the next terms

a2-a1 = 6 – 2 = 4

a3-a2 = 10 –6 = 4

a4-a3 = 14 – 10 = 4

Since the differences are common.

Hence the given terms are in A.P.

Problem : Find the value of k for which the given series is in A.P. 4, k –1 , 12

Solution : Given A.P. is 4, k –1 , 12…..

If series is A.P. then the differences will be

common.

d1 = d1

a2 – a1 = a3 – a2

k – 1 – 4 = 12 – (k – 1)

k – 5 = 12 – k + 1

k + k = 12 + 1 + 5

2 k = 18 or k = 9

The sum of n terms, we find as,

Sum = n X [(first term + last term) / 2] Now last term will be = a + (n-1) d

Therefore,

Sum(Sn) =n X [{a + a + (n-1) d } /2 ]

= n/2 [ 2a + (n+1)d]

DERIVATION

The sum to n terms is given by: Sn = a + (a + d) + (a + 2d) + … + (a + (n – 1)d)     (1)

If we write this out backwards, we get:Sn = (a + (n – 1)d) + (a + (n – 2)d) + … +a  (2)

          

Now let’s add (1) and (2):

2Sn = [2a + (n – 1)d] + [2a + (n – 1)d] + ………… + [2a + (n – 1)d]

So, Sn = n/2 [2a + (n – 1)d]

Problem . Find number of terms of A.P. 100, 105, 110, 115,,………………500Solution.

First term is a = 100 , an = 500

Common difference is d = 105 -100 = 5

nth term is an = a + (n-1)d

500 = 100 + (n-1)5

500 - 100 = 5(n – 1)

400 = 5(n – 1)

5(n – 1) = 400

5(n – 1) = 400

n – 1 = 400/5

n - 1 = 80

n = 80 + 1

n = 81

Hence the no. of terms are

81.

Problem . Find the sum of 30 terms of given A.P. ,12 , 20 , 28 , 36………

Solution : Given A.P. is 12 , 20, 28 , 36

Its first term is a = 12

Common difference is d = 20 – 12 = 8

The sum to n terms of an arithmetic progression

Sn = n/2 [ 2a + (n - 1)d ]

= ½ x 30 [ 2x 12 + (30-1)x 8]

= 15 [ 24 + 29 x8]

= 15[24 + 232]

= 15 x 246

= 3690

THE SUM OF TERMS IS 3690

Problem . Find the sum of terms in given A.P.

2 , 4 , 6 , 8 , ……………… 200

Solution: Its first term is a = 2

Common difference is d = 4 – 2 = 2

nth term is an = a + (n-1)d

200 = 2 + (n-1)2

200 - 2 = 2(n – 1)

2(n – 1) = 198

n – 1 = 99, n = 100

The sum to n terms of an arithmetic progression

Sn = n/2[ 2a + (n - 1)d ]

S100 = 100/2 [ 2x 2 + (100-1)x 2]

= 50 [ 4 + 198]

= 50[202]

= 10100

The difference between two terms of an AP can be formulated as below:-

nth term – kth term= t(n) – t(k) = {a + (n-1)d} – { a + (k-1) d } = a + nd – d – a – kd + d = nd – kd Hence, t(n) – t(k) = (n – k) d